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Jun 29, 2018
06/18

by
Vladimir García-Morales

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A simple discontinuous map is proposed as a generic model for nonlinear dynamical systems. The orbit of the map admits exact solutions for wide regions in parameter space and the method employed (digit manipulation) allows the mathematical design of useful signals, such as regular or aperiodic oscillations with specific waveforms, the construction of complex attractors with nontrivial properties as well as the coexistence of different basins of attraction in phase space with different...

Topics: Chaotic Dynamics, Nonlinear Sciences

Source: http://arxiv.org/abs/1601.03017

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Jun 29, 2018
06/18

by
Vladimir García-Morales

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A minimalistic model for chimera states is presented. The model is a cellular automaton (CA) which depends on only one adjustable parameter, the range of the nonlocal coupling, and is built from elementary cellular automata and the majority (voting) rule. This suggests the universality of chimera-like behavior from a new point of view: Already simple CA rules based on the majority rule exhibit this behavior. After a short transient, we find chimera states for arbitrary initial conditions, the...

Topics: Cellular Automata and Lattice Gases, Pattern Formation and Solitons, Nonlinear Sciences, Chaotic...

Source: http://arxiv.org/abs/1602.03799

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Jun 30, 2018
06/18

by
Vladimir García-Morales

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It is shown that any two cellular automata (CA) in rule space can be connected by a continuous path parameterized by a real number $\kappa \in (0, \infty)$, each point in the path corresponding to a coupled map lattice (CML). In the limits $\kappa \to 0$ and $\kappa \to \infty$ the CML becomes each of the two CA entering in the connection. A mean-field, reduced model is obtained from the connection and allows to gain insight in those parameter regimes at intermediate $\kappa$ where the dynamics...

Topics: Pattern Formation and Solitons, Cellular Automata and Lattice Gases, Mathematical Physics,...

Source: http://arxiv.org/abs/1701.01281

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Jul 20, 2013
07/13

by
Vladimir Garcia-Morales

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A universal map is derived for all deterministic 1D cellular automata (CA) containing no freely adjustable parameters. The map can be extended to an arbitrary number of dimensions and topologies and its invariances allow to classify all CA rules into equivalence classes. Complexity in 1D systems is then shown to emerge from the weak symmetry breaking of the addition modulo an integer number p. The latter symmetry is possessed by certain rules that produce Pascal simplices in their time...

Source: http://arxiv.org/abs/1203.3939v1

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Jun 27, 2018
06/18

by
Vladimir Garcia-Morales

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A mathematical method for constructing fractal curves and surfaces, termed the $p\lambda n$ fractal decomposition, is presented. It allows any function to be split into a finite set of fractal discontinuous functions whose sum is equal everywhere to the original function. Thus, the method is specially suited for constructing families of fractal objects arising from a conserved physical quantity, the decomposition yielding an exact partition of the quantity in question. Most prominent classes of...

Topics: Mathematical Physics, Statistical Mechanics, Mathematics, Condensed Matter

Source: http://arxiv.org/abs/1505.02547

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Jun 27, 2018
06/18

by
Vladimir Garcia-Morales

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A simple mathematical expression for the universal map for cellular automata is found in closed form with the help of a digit function, whose most basic properties are established. This result is found after proving a theorem on the composition of functions on finite sets. The expression (and the technique used to obtain it) opens the possibility of gaining mathematical insight in any cellular automaton rule since it constitutes at the same time a simple and fast algorithm to implement any such...

Topics: Cellular Automata and Lattice Gases, Nonlinear Sciences

Source: http://arxiv.org/abs/1505.02543

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Jun 29, 2018
06/18

by
Vladimir García-Morales

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We introduce $\mathcal{B}_{\kappa}$-embeddings, nonlinear mathematical structures that connect, through smooth paths parameterized by $\kappa$, a finite or denumerable set of objects at $\kappa=0$ (e.g. numbers, functions, vectors, coefficients of a generating function...) to their ordinary sum at $\kappa \to \infty$. We show that $\mathcal{B}_{\kappa}$-embeddings can be used to design nonlinear irreversible processes through this connection. A number of examples of increasing complexity are...

Topics: Pattern Formation and Solitons, Mathematical Physics, Nonlinear Sciences, Adaptation and...

Source: http://arxiv.org/abs/1607.02889

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Jun 29, 2018
06/18

by
Vladimir García-Morales

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The union of a collection of $n$ sets is generally expressed in terms of a characteristic (indicator) function that contains $2^{n}-1$ terms. In this article, a much simpler expression is found that requires the evaluation of $n$ terms only. This leads to a major simplification of any normal form involving characteristic functions of sets. The formula can be useful in recognizing inclusion-exclusion patterns of combinatorial problems.

Topics: General Mathematics, Mathematics

Source: http://arxiv.org/abs/1608.00861

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Jun 30, 2018
06/18

by
Vladimir Garcia-Morales

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A new variational method, the principle of least radix economy, is formulated. The mathematical and physical relevance of the radix economy, also called digit capacity, is established, showing how physical laws can be derived from this concept in a unified way. The principle reinterprets and generalizes the principle of least action yielding two classes of physical solutions: least action paths and quantum wavefunctions. A new physical foundation of the Hilbert space of quantum mechanics is...

Topics: Physics, Quantum Physics, General Physics

Source: http://arxiv.org/abs/1401.0963

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Jun 27, 2018
06/18

by
Vladimir Garcia-Morales

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By means of a digit function that has been introduced in a recent formulation of classical and quantum mechanics, we provide a new construction of some infinite families of finite groups, both abelian and nonabelian, of importance for theoretical, atomic and molecular physics. Our construction is not based on algebraic relationships satisfied by generators, but in establishing the appropriate law of composition that induces the group structure on a finite set of nonnegative integers (the...

Topics: Group Theory, Mathematical Physics, Mathematics, High Energy Physics - Theory

Source: http://arxiv.org/abs/1505.04528

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Jun 29, 2018
06/18

by
Vladimir García-Morales

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A general mathematical method is presented for the systematic construction of coupled map lattices (CMLs) out of deterministic cellular automata (CAs). The entire CA rule space is addressed by means of a universal map for CAs that we have recently derived and that is not dependent on any freely adjustable parameters. The CMLs thus constructed are termed real-valued deterministic cellular automata (RDCA) and encompass all deterministic CAs in rule space in the asymptotic limit $\kappa \to 0$ of...

Topics: Cellular Automata and Lattice Gases, Pattern Formation and Solitons, Nonlinear Sciences, Chaotic...

Source: http://arxiv.org/abs/1602.00289

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Jun 28, 2018
06/18

by
Vladimir Garcia-Morales

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Fractal surfaces ('patchwork quilts') are shown to arise under most general circumstances involving simple bitwise operations between real numbers. A theory is presented for all deterministic bitwise operations on a finite alphabet. It is shown that these models give rise to a roughness exponent $H$ that shapes the resulting spatial patterns, larger values of the exponent leading to coarser surfaces.

Topics: Computing Research Repository, Other Computer Science

Source: http://arxiv.org/abs/1507.01444

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Jun 28, 2018
06/18

by
Vladimir García-Morales

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A new class of deterministic dynamical systems, termed semipredictable dynamical systems, is presented. The spatiotemporal evolution of these systems have both predictable and unpredictable traits, as found in natural complex systems. We prove a general result: The dynamics of any deterministic nonlinear cellular automaton (CA) with $p$ possible dynamical states can be decomposed at each instant of time in a superposition of $N$ layers involving $p_{0}$, $p_{1}$,... $p_{N-1}$ dynamical states...

Topics: Cellular Automata and Lattice Gases, Chaotic Dynamics, Mathematical Physics, Mathematics, Nonlinear...

Source: http://arxiv.org/abs/1507.08455

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Jun 29, 2018
06/18

by
Vladimir García-Morales

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We present a diagrammatic method to build up sophisticated cellular automata (CAs) as models of complex physical systems. The diagrams complement the mathematical approach to CA modeling, whose details are also presented here, and allow CAs in rule space to be classified according to their hierarchy of layers. Since the method is valid for any discrete operator and only depends on the alphabet size, the resulting conclusions, of general validity, apply to CAs in any dimension or order in time,...

Topics: Nonlinear Sciences, Cellular Automata and Lattice Gases, Mathematics, Pattern Formation and...

Source: http://arxiv.org/abs/1605.06937

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Jun 30, 2018
06/18

by
Vladimir García-Morales

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It is shown that characteristic functions of classical `crisp' sets can be made fuzzy by means of a $\mathcal{B}_{\kappa}$-function that we have recently introduced, and where the fuzziness parameter $\kappa \in \mathbb{R}$ controls how much a fuzzy set deviates from the crisp set obtained in the limit $\kappa \to 0$. A theorem is established that yields novel expressions for the union, negation, and intersection of both classical and fuzzy sets. As an application, we establish a theorem on the...

Topics: Chaotic Dynamics, Nonlinear Sciences, Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1704.00676

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Jun 30, 2018
06/18

by
Lennart Schmidt; Konrad Schönleber; Vladimir García-Morales; Katharina Krischer

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The photoelectrodissolution of n-type silicon constitutes a convenient model system to study the nonlinear dynamics of oscillatory media. On the silicon surface, a silicon oxide layer forms. In the lateral direction, the thickness of this layer is not uniform. Rather, several spatio-temporal patterns in the oxide layer emerge spontaneously, ranging from cluster patterns and turbulence to quite peculiar dynamics like chimera states. Introducing a nonlinear global coupling in the complex...

Topics: Nonlinear Sciences, Pattern Formation and Solitons, Adaptation and Self-Organizing Systems

Source: http://arxiv.org/abs/1405.0414